Congruences of fork extensions of slim semimodular lattices

For a slim, planar, semimodular lattice $L$ and covering square~$S$, G.~Cz\'edli and E.\,T.~Schmidt introduced the fork extension, $L[S]$, which is also a slim, planar, semimodular lattice. We investigate when a congruence of $L$ extends to $L[S]$. We introduce a join-irreducible congruence $\boldsymbol{\gamma}(S)$ of $L[S]$. We determine when it is new, in the sense that it is not generated by a join-irreducible congruence of $L$. When it is new, we describe the congruence $\boldsymbol{\gamma}(S)$ in great detail. The main result follows: \emph{In the order of join-irreducible congruences of a slim, planar, semimodular lattice $L$, the congruence $\boldsymbol{\gamma}(S)$ has \emph{at most two covers.}}Comment: arXiv admin note: substantial text overlap with arXiv:1307.6126, arXiv:1307.077

Congruences and trajectories in planar semimodular lattices

A 1955 result of J.~Jakub\'i k states that for the prime intervals \fp and \fq of a finite lattice, \con{\fp} \geq \con{\fq} if{}f \fp is congruence-projective to~\fq (\emph{via} intervals of arbitrary size). The problem is how to determine whether \con{\fp} \geq \con{\fq} involving only prime intervals. Two recent papers approached this problem in different ways. G. Cz\'edli's used trajectories for slim rectangular lattices---a special subclass of slim, planar, semimodular lattices. I used the concept of prime-projectivity for arbitrary finite lattices. In this note I show how my approach can be used to generalize Cz\'edli's result to arbitrary slim, planar, semimodular lattices.Comment: The topic of this paper was subsumed by the paper Congruences and prime-perspectivities in finite lattice

Congruences of the fork extensions. I. The Congruence Extension Property

For a slim, planar, semimodular lattice, G. Cz\'edli and E.\,T. Schmidt introduced the fork extension in 2012. In this note we prove that the fork extension has the Congruence Extension Property. This paper has been merged with Part II, under the title Congruences of fork extensions of slim semimodular lattices, see arXiv: 1307.8404Comment: arXiv admin note: substantial text overlap with arXiv:1307.840

A technical lemma for congruences of finite lattices

The classical Technical Lemma for congruences is not difficult to prove but it is very efficient in its applications. We present here a Technical Lemma for congruences on \emph{finite lattices}. This is not difficult to prove either but it has already has proved its usefulness in some applications

Congruences of the fork extensions. II. The congruence gamma

G. Cz\'edli and E.\,T. Schmidt introduced in 2012 the fork extension. Continuing from Part I, we investigate the congruences of a fork extension. This paper has been merged with Part I, under the title Congruences of fork extensions of slim semimodular lattices, see arXiv: 1307.8404Comment: arXiv admin note: substantial text overlap with arXiv:1307.840

Homomorphisms and principal congruences of bounded lattices

Two years ago, I characterized the order \Princl L of principal congruences of a bounded lattice $L$ as a bounded order. If $K$ and $L$ are bounded lattices and \gf is a \zo homomorphism of $K$ into~$L$, then there is a natural isotone \zo-map \gf_{\Hom} from \Princl K into \Princl L. We prove the converse: For bounded orders $P$ and $Q$ and an isotone \zo map \gy of $P$ into $Q$, we represent $P$ and $Q$ as \Princl K and \Princl L for bounded lattices $K$ and $L$ with a \zo homomorphism \gf of $K$ into $L$, so that \gy is represented as \gf_{\Hom}

Tensor products and transferability of semilattices

In general, the tensor product, $A\otimes B$, of the lattices A and B with zero is not a lattice (it is only a join-semilattice with zero). If $A \otimes B$ is a capped tensor product, then $A \otimes B$ is a lattice (the converse is not known). In this paper, we investigate lattices A with zero enjoying the property that $A \otimes B$ is a capped tensor product, for every lattice B with zero; we shall call such lattices amenable. The first author introduced in 1966 the concept of a sharply transferable lattice. In 1972, H. Gaskill [5] defined, similarly, sharply transferable semilattices, and characterized them by a very effective condition (T). We prove that a finite lattice A is amenable iff it is sharply transferable as a join-semilattice. For a general lattice A with zero, we obtain the result: A is amenable iff A is locally finite and every finite sublattice of A is transferable as a join-semilattice. This yields, for example, that a finite lattice A is amenable iff $A\otimes F(3)$ is a lattice iff A satisfies (T), with respect to \jj. In particular, $M3 \otimes F(3)$ is not a lattice. This solves a problem raised by R. W. Quackenbush in 1985 whether the tensor product of lattices with zero is always a lattice

Tensor products of semilattices with zero, revisited

Let A and B be lattices with zero. The classical tensor product, $A\otimes B$, of A and B as join-semilattices with zero is a join-semilattice with zero; it is, in general, not a lattice. We define a very natural condition: $A \otimes B$ is capped (that is, every element is a finite union of pure tensors) under which the tensor product is always a lattice. Let Conc L denote the join-semilattice with zero of compact congruences of a lattice L. Our main result is that the following isomorphism holds for any capped tensor product: $Conc A\otimes Conc B \cong Conc(A \otimes B)$. This generalizes from finite lattices to arbitrary lattices the main result of a joint paper by the first author, H. Lakser, and R. W. Quackenbush

Proper congruence-preserving extensions of lattices

We prove that every lattice with more than one element has a proper congruence-preserving extension

Flat semilattices

Let $A$, $B$, and $S$ be (v,0)-semilattices and let $f: A\to B$ be a (v,0)-embedding. Then the canonical map, f \otimes \id\_S, of the tensor product $A \otimes S$ into the tensor product $B \otimes S$ is not necessarily an embedding. The (v,0)-semilattice $S$ is flat, if for every embedding $f : A\to B$, the canonical map f\otimes\id is an embedding. We prove that a (v,0)-semilattice $S$ is flat if and only if it is distributive